Finite Group Databasebeta

← #125272

Subgroups — #125272

Order 512.

433 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33129C211yesmaximal, characteristic
2C2×#1374C211yesmaximal, characteristic, complemented
3#7875C211yesmaximal, characteristic, complemented
4#7121C212yesmaximal, complemented
5#7112C212yesmaximal, complemented
6C23×C16—21no—
7C2×#466C411yescharacteristic
8C2×#432—21no—
9C22×C4×C8C2211yescharacteristic
10#2854—21no—
11#1970C411yescharacteristic
12#1970C2211yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1934—21no—
16#1934—21no—
17#1893—21no—
18#1808C411yescharacteristic
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×D8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123—41no—
27C2×#121—21no—
28C22×C42D411yescharacteristic
29C22×C16—24no—
30C22×C16—22no—
31C22×C16—42no—
32C22×C16—42no—
33#466—42no—
34#432—24no—
35#432—42no—
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×C4⋊C8—22no—
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×C22⋊C8—41no—
44C2×#97—41no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#390—21no—
51#390—21no—
52#389—21no—
53#389—21no—
54#388—41no—
55#388—41no—
56#386—41no—
57#380—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×D4—21no—
62C22×D4—41no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4SD1611yescharacteristic
66C2×D8—22no—
67C2×D8—41no—
68C2×D8—41no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8—42no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#123—84no—
82#121—42no—
83#121—42no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42D811yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C16—24no—
102C2×C16—48no—
103C2×C16—42no—
104C2×C16—42no—
105#107—22no—
106#107—41no—
107#107—41no—
108C4⋊C8—24no—
109C4⋊C8—42no—
110#102—42no—
111#102—42no—
112#102—41no—
113#102—41no—
114C22⋊C8—44no—
115#97—44no—
116C4×C8—41no—
117C4×C8—41no—
118C4×C8—41no—
119C4×C8—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123C24—41no—
124C24#10211yescharacteristic
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—82no—
132C2×D4—82no—
133C2×D4—82no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4#9912yes—
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10412yes—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×SD1611yescharacteristic
155D8—82no—
156C2×C8—24no—
157C2×C8—24no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—82no—
170C2×C8—82no—
171C2×C8#10412yes—
172C2×C8#13112yes—
173C2×C8AGL(1,ℤ/8)11yescharacteristic
174C2×C8#13711yescharacteristic
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C22⋊C4—82no—
187C22⋊C4—82no—
188C22⋊C4—82no—
189C42—22no—
190C42—22no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C16—44no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—81no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23—81no—
207C23#32712yes—
208C23#35912yes—
209C23#36112yes—
210C23C2×#10211yescharacteristic
211D4—81no—
212D4—81no—
213D4—81no—
214D4—81no—
215D4—164no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4#32712yes—
245C2×C4#40811yescharacteristic
246C2×C4C2×#9911yescharacteristic
247C2×C4#41311yescharacteristic
248C2×C4#41711yescharacteristic
249C2×C4C2×#10411yescharacteristic
250C2×C4C2×#10411yescharacteristic
251C2×C4#50812yes—
252C8—22no—
253C8—21no—
254C8—22no—
255C8—21no—
256C8—41no—
257C8—84no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—161no—
265C22—161no—
266C22—161no—
267C22—161no—
268C22—161no—
269C22—161no—
270C22—161no—
271C22#134914yes—
272C22#149211yescharacteristic
273C22C2×#32711yescharacteristic
274C22#154711yescharacteristic
275C22#213811yescharacteristic
276C22C2×#35911yescharacteristic
277C22#214511yescharacteristic
278C22C2×#36111yescharacteristic
279C4—22no—
280C4—22no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—161no—
285C4—161no—
286C4—161no—
287C4#146811yescharacteristic
288C4#148311yescharacteristic
289C4#150411yescharacteristic
290C4#152411yescharacteristic
291C2—24no—
292C2—321no—
293C2#767811yescharacteristic
294C2#1167311yescharacteristic
295C2#1167811yescharacteristic
296C2#1170311yescharacteristic
297C2C2×#134911yescharacteristic
298C2#1197811yescharacteristic
299C2#1198211yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1374C2semidirect
3. #7875C2semidirect
4. #7121C2semidirect
5. #7112C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331292
C2×#13742
#78752
#71212
#71122